Video Transcript
Line segment 𝐴𝐵 is a uniform
rod of length four centimeters and mass four kilograms. A mass of magnitude five
kilograms is fixed at 𝐴, and another mass of magnitude one kilogram is fixed at
𝐵. Find the distance from the
center of gravity of the system to 𝐴.
The first piece of information
we’re given is that the rod, which is represented by a line segment 𝐴𝐵, is
uniform. Now, of course, we know that a
uniform rod has a constant density. So the center of mass of this
rod is located at its midpoint. Let’s use this information and
the information about the masses that are fixed at 𝐴 and 𝐵 to represent this
in a diagram. The next thing that we can do
is draw a coordinate line to indicate the position of these masses with respect
to the coordinate plane. Let’s imagine then that point
𝐴 is at the origin and the line segment 𝐴𝐵 lies along the 𝑥-axis as
shown.
Since the length of the rod is
four centimeters, if we let centimeters be our length unit, then point 𝐵 must
have coordinates four, zero. Then the midpoint of the line
segment 𝐴𝐵 is found by calculating the sum of the individual coordinates and
dividing by two. So that’s zero plus four
divided by two and zero plus zero divided by two, which gives us the coordinates
two, zero. Now, in fact, we’re really only
interested in the value of the 𝑥-coordinates. But we’ve completed the value
of the 𝑦-coordinates just for completeness.
Next, we recall that if a
system of particles has mass 𝑚 sub 𝑖 at coordinates 𝑥 sub 𝑖, 𝑦 sub 𝑖, then
the 𝑥- and 𝑦-coordinates of the center of mass are as shown. Since the three centers of mass
that we’re interested in lie along the 𝑥-axis, we’re just going to use the
calculation for the 𝑥-coordinates. Let’s begin by working out the
sum of the products of 𝑚 sub 𝑖 and 𝑥 sub 𝑖. The mass of particle 𝐴 is five
kilograms, and its 𝑥-coordinate is zero. The mass of the uniform rod is
four kilograms, and its center of mass is located at the point two, zero. And finally, the mass of the
weight at point 𝐵 is one kilogram, and it’s located four units along the
𝑥-axis. And so the sum of 𝑚 sub 𝑖
times 𝑥 sub 𝑖 is 12 or 12 kilogram centimeters.
The denominator of the fraction
is the sum of 𝑚 sub 𝑖. And that’s simply the sum of
the masses of all the objects in the system. It’s five plus four plus one,
which is equal to 10, or 10 kilograms. The center of mass is the
quotient of these, so it’s 12 divided by 10, or 1.2. And so the 𝑥-coordinate of the
center of mass is 1.2. And, in fact, we know that that
measurement is in centimeters. Since this point lies on the
𝑥-axis and point 𝐴 lies at the origin, the distance from the center of mass or
center of gravity of the system to 𝐴 is 1.2 centimeters.
